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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
28/1999

Convex integration with constraints and applications to phase transitions and partial differential equations

Stefan Müller and Vladimír Šverák

Abstract

We study solutions first order partial differential relations $Du \in K$, where $u: \Omega R^n \to R^m$ is a Lipschitz map and $K$ is a bounded set in $m \times n$ matrices, and extend Gromov's theory of convex integration in two ways. First, we allow for additional constraints on the minors of $Du$ and second we replace Gromov's $P$-convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of `wild' solutions to elliptic systems. Our work was originally motivated by questions in the analysis of crystal microstructure and we establish the existence of a wide class of solutions to the two-well problem in the theory of martensite.

Received:
02.05.99
Published:
02.05.99

Related publications

inJournal
1999 Repository Open Access
Stefan Müller and Vladimír Šverák

Convex integration with constraints and applications to phase transitions and partial differential equations

In: Journal of the European Mathematical Society, 1 (1999) 4, pp. 393-422